A thin elastic wire is placed between rigid supports. A fluid flows past the wire, and it is desired to study the static deflection. \(\delta,\) at the center of the wire due to the fluid drag. Assume that $$\delta=f(\ell, d, \rho, \mu, V, E)$$ where \(\ell\) is the wire length, \(d\) the wire diameter, \(\rho\) the fluid density, \(\mu\) the fluid viscosity, \(V\) the fluid velocity, ind \(E\) the modulus of elasticity of the wire material. Develop a suitable set of pi terms for this problem.

Firstly, identify the parameters involved in the deflection of the wire and write their dimensions. These include:\n Length of the wire, \(\ell\) [L]\n Modulus of elasticity, E [ML^-1T^-2]\n Wire diameter, d [L]\n Fluid density, \(\rho\) [ML^-3]\n Fluid viscosity, \(\mu\) [ML^-1T^-1]\n Fluid velocity, V [LT^-1]

Choose a set of repeating (or recurring) variables which should have the following properties:\n The chosen repeating variables should contain all the basic dimensions i.e., M, L and T.\n The variable to be modeled, in this case \(\delta\), should not be a repeating variable.\n \nIn this case, \(\ell\), \(\rho\), and V are chosen as repeating variables since they contain all the dimensions M, L and T.

Form the pi terms by equating the dimensions on the left and right side of the Pi term equation. Each Pi term is a dimensionless group formed by taking the dependent variable and multiplying it by the repeating variables each to an arbitrary power.\n\nPi Term 1: To avoid \(\delta\) from being in more than one Pi term, choose it for the first Pi term. The Pi term can be expressed as: \(\Pi_{1} = \delta \cdot \ell^{a} \cdot \rho^{b} \cdot V^{c}\) and equating the dimensions of both sides of the equation, powers a, b, c are determined.\n\nPi Term 2: Chosen variable is \(d\). The Pi term can be expressed as: \(\Pi_{2} = d \cdot \ell^{a} \cdot \rho^{b} \cdot V^{c}\) and by equating the dimensions, \(a=-1, b=0, c=0\).\n\nPi Term 3: Chosen variable is \(\mu\). The Pi term can be expressed as: \(\Pi_{3} = \mu \cdot \ell^{a} \cdot \rho^{b} \cdot V^{c}\) and by equating the dimensions, \(a=-1, b=-1, c=1\).\n\nPi Term 4: Chosen variable is \(E\). The Pi term can be expressed as: \(\Pi_{4} = E \cdot \ell^{a} \cdot \rho^{b} \cdot V^{c}\) and by equating the dimensions on both sides of the equation, \(a=0, b=1, c=0\). After calculating the unknowns, the Pi terms simplify.

After simplifying, the final dimensionless Pi terms would be:\n\n\(\Pi_{1} = \delta / \ell\), \(\Pi_{2} = d / \ell\), \(\Pi_{3} = \mu / (\rho V \ell)\), \(\Pi_{4} = E / \rho\)